Let K be a field of characteristic 0 and let n be a natural number. Let Γ be a subgroup of the multiplicative group (K *

w 1. IntroductionSeveral classes of diophantine equations, such as the Thue-Mahler equation and certain generalisations of the Ramanujan-Nagell equation, can be reduced to certain linear equations in two S-units. Here S is a finite set of equivalence classes of valuations on a given algebraic number field K and an S-unit is an element eeK with the property that the only valuations on K which assume a value 4= 1 for e belong to equivalence classes from S. In this section we shall state a general result on the number of solutions of linear equations in two Sunits. Before we can state it, we have to introduce some notions on valuations. In w we shall discuss the consequences of our general result for generalisations of the Ramanujan-Nagell equation and in w167 3-4 we shall deal with the Thue-Mahler equation.Let K be an algebraic number field. By a prime on K we shall mean an equivalence class of non-trivial valuations on K. Usually we shall denote primes on Q by the letter p, on a given algebraic number field K by v and on an extension of K by V. The completion of K at the prime v is denoted by K v.We distinguish between finite primes, containing non-archimedean valuations, and infinite primes, containing archimedean valuations.

Abstract:A new protocol is presented that allows A to convince B that she knows a solution to the Discrete Log Problem-i.e. that she knows an x such that dc E /3 (mod N ) holdswithout revealing anydung about x to B. Protocols are given both for N prime and for N composite.show possession of: -We also give protocols for extensions of the Discrete Log problem allowing A to multiple discrete logarithms to the same base at the same time, i.e. knowing several discrete logarithms to different bases at the same time, i.e. knowing X I , . . . ,XK such that the product a;'a;' . . . a? ~8 ; a discrete logarithm that is the simultaneous solution of several different instances, i.e. knowing x such that a t z /31, . . . , a% E PK.We can prove that the sequential versions of these protocols do not reveal any "knowledge" about the discrete logarithm(s) in a well-defined sense, provided that A knows (a multiple of) the order of a.

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Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.